Studying the temporal dynamics of bistable perception can be useful for understanding neural mechanisms underlying the phenomenon. We take a closer look at those temporal dynamics, using data from four different ambiguous stimuli. We focus our analyses on two recurrent themes in bistable perception literature. First, we address the question whether percept durations follow a gamma distribution, as is commonly assumed. We conclude that this assumption is not justified by the gamma distribution’s approximate resemblance to distributions of percept durations. We instead present two straightforward distributions of reciprocal percept durations (i.e., rates) that both easily surpass the classic gamma distribution in terms of resemblance to empirical data. Second, we compare the distributions arising from binocular rivalry with those from other forms of bistable perception. Parallels in temporal dynamics between those classes of stimuli are often mentioned as an indication of a similar neural basis, but have never been studied in detail. Our results demonstrate that the distributions arising from binocular rivalry and other forms of bistable perception are indeed similar up to a high level of detail.

*n*) is the canonical continuous extension of (

*n*− 1)!, which itself is of course only defined for natural

*n*. The parameters

*k*and

*λ*in the equation are usually called the shape parameter and the scale parameter, respectively.

^{2}probability lower than 1%, and De Marco and colleagues mentioned two alternative theoretical distributions to fit equally well as the gamma distribution, although less favorable in the light of parsimony.) The two remaining studies (Cogan, 1973; Zhou, Gao, White, Merk, & Yao, 2004) show an unacceptable fit quality for the gamma distribution: In both cases more than half of the fitted distributions should be rejected at the 5% significance level. As a point in favor of the gamma distribution, it should be mentioned that on the basis of the above studies, one cannot identify an alternative distribution with better fit performance. Although Zhou et al. (2004) did show the lognormal distribution to fit their data better than the gamma distribution, Cogan (1973) rejected the lognormal distribution as an acceptable fit to her data (note that lognormal distributed percept durations were also proposed by Lehky, 1995).

*perceptual alternation rates*(i.e., reciprocal percept durations), being more directly related to underlying neural processes than distributions of percept durations, could give insight into neural processes. This especially makes sense in light of the idea that the numerous phenomenological similarities between saccadic search and bistable perception are not merely coincidental, but instead reflect a more fundamental link between the two (Leopold & Logothetis, 1999). Completing the analogy between rate models in saccadic search and in bistable perception, we therefore suggest a decision signal that starts rising at the beginning of a dominance period of percept

*A*, and causes an alternation to percept

*B*as soon as it reaches its threshold, so that characteristics of the rate of information accumulation are reflected in distributions of perceptual alternation rates.

*S*and

*F*of the beta′ distribution are determined by the thresholds for switching percepts and maintaining the current one, respectively. The parameter

*r*is proportional to the observed alternation rate via a constant

*R*: rate =

*Rr*. In the present work, we fix

*R*at 1 s

^{−1}, performing our fits with a two-parameter distribution. The main reason for this is that the full three-parameter version caused divergence in our fitting algorithm, while much of the distribution’s flexibility can already be achieved by varying only the two remaining parameters. It should be kept in mind however that removing this constraint on the third parameter would provide room for an improved fit quality compared to what we present here. A complete derivation of the beta′ distribution from the model assumptions is given by Van den Berg and Van Loon (in press).

*A*has a beta′ distribution (with parameters

*S*and

*F*), then 1/

*A*has a beta′ distribution, too (with parameters

*F*and

*S*). The important implication here is that the beta′ distribution will by definition fit our distributions of percept durations equally well as our distributions of alternation rates, thereby in a sense bridging the gap between the two approaches. Figure 2 gives an impression of the overall shape of the probability density functions (PDFs) and the cumulative distribution functions (CDFs) associated with the gamma duration, the gamma rate, and the beta′ rate model. It shows both the overall similarity between them and the more subtle aspects in which they differ.

- Orthogonal grating stimulus. Orthogonal grating rivalry is a classical form of binocular rivalry, where orthogonal gratings are presented to the two eyes. In our case the stimulus consisted of four parallel lines at 45 deg with the vertical that had orthogonal orientations in the two eyes.
- Bistable slant stimulus. This is a stimulus first systematically studied by Van Ee, Van Dam, and Erkelens (2002), consisting of a trapezoid that is viewed binocularly. Bistability arises from the fact that linear perspective and binocular disparity specify opposite slants (e.g., in Figure 1B perspective information corresponds to a rectangle seen with the right side in front, whereas disparity information corresponds to a trapezoid seen with the left side in front).
- House-face stimulus. This is a stimulus developed by Tong, Nakayama, Vaughan, and Kanwisher (1998). Like the orthogonal gratings described above, it gives rise to a form of binocular rivalry. In house-face rivalry, however, the conflicting images are not orthogonal lines but pictures of a house and a face.
- Necker cube stimulus. This is arguably the best known ambiguous stimulus. It is an image that can be interpreted as a cube seen from either of two viewpoints.

*SSE*. For the fits we obtained in this manner, we calculated the fit quality as

*SSE*/(

*n*− 2) (where

*n*is the number of data points in a given distribution), but also using the Kolmogorov-Smirnov test for goodness of fit. The Kolmogorov-Smirnov test involves the largest overall deviation between empirical and fitted distribution, and the associated probability

*pKS*. In addition to these least squares fits, we performed maximum likelihood fits on the same data, providing an alternative estimate of the best-fitting parameter values, as well as of the fit quality: the likelihood L. Note that likelihood fits by definition involve PDFs, whereas for the least squares fits we used continuity corrected CDFs, for reasons of robustness and objectivity (contrary to the PDF, the CDF does not involve an arbitrary bin size). All fitting algorithms were implemented in the software package Scilab (http://scilabsoft.inria.fr/).

*p*level that one chooses. An advantage of this presentation method is that it summarizes the acceptance of fits, without pinpointing the critical

*pKS*level in advance, revealing that both rate distributions have a higher acceptance than the gamma duration distribution at any critical

*p*level. For instance, in case one should decide to choose a level of 0.1 (vertical line in Figure 5), fractions of 0.92, 0.86, and 0.69 of the gamma rate, beta′ rate, and gamma duration fits pass, respectively.

*manner*in which they deviate. In other words, two empirical distributions might fit a given model equally well, but if one of them consistently undershoots where the other overshoots, this similar fit quality does not reflect a similar shape. We therefore performed an additional analysis in which we compared the distributions’ shapes in a more direct way, by looking at the nature of the deviations between empirical and fitted distributions. For this purpose, we calculated for all three fits the mean fit residual as a function of position along the CDF, averaging over all 48 distributions associated with a particular stimulus. Because every CDF spans a different domain of rates (or durations), the data had to be standardized to combine them and cal culate this average. A simple but effective way to achieve this is to consider the residuals as a function of the value of the fitted CDF, instead of as a function of the rate (or duration). Our procedure is illustrated in Figure 7.

*A*of an empirical rate distribution, the part containing low rates, can be described by one truncated Gaussian distribution Φ

_{A}, and the remaining part

*B*by another one Φ

_{B}(Carpenter & Williams, 1995; Reddi & Carpenter, 2000). The raw data are plotted on probability paper, so the distinction between data section

*A*and

*B*can be made by eye, after which the parameters of Φ

_{A}and Φ

_{B}can be estimated (see Figure 10, left panel). If we want to compare the fit quality of the dual Gaussian distribution to that of the ones we investigated, this approach will not do, mainly because it involves visual inspection. Because it does seem that the combination of two Gaussians describes rates of saccadic eye movements well, we adjusted the dual Gaussian distribution to investigate its fit quality to our data on alternation rates. According to this adaptation, the probability density function at rate

*r*is given by .

*ϕ*is a Gaussian distribution. Note that instead of fitting two truncated Gaussians to separate parts of the empirical distribution, we fit a mix of two Gaussians to the entire distribution, and introduce a free scaling parameter

*a*to ensure a total cumulative probability of 1. This mixed Gaussian distribution was introduced over a century ago (Pearson, 1894), and similar mixed distributions are presently used in numerous fields of research (McLachlan & Peel, 2000). This dual Gaussian model produced excellent fits to our data (all were accepted at a critical

*pKS*level of 0.1, after removal of nonconvergent fits), but a direct comparison to the other models is not possible because these contain only two free parameters instead of five (the dual Gaussian distribution is over-parameterized, as witnessed by the convergence problems we experienced). In addition, it is worth mentioning that our adapted dual Gaussian fits often reach quite different results than the conventional ones (see Figure 10). A final conceptual problem one might have with these fits is that both in the original form and in our adaptation, they allow for the occurrence of negative alternation rates.

*k*reflects the number of Poisson ticks after which a perceptual transition takes place (see Introduction). On the basis of the model, one might therefore predict that this shape parameter should be a natural number. What Murata and coworkers showed is that the gamma duration distributions they fitted to their data indeed had shape parameters that grouped around natural numbers. It would seem that this result cannot be explained unless by accepting the idea of a Poisson clock and the associated gamma duration distribution, but we see two reasons why such a conclusion is premature. First of all, Murata and coworkers did not support their claim with a statistical analysis. A straightforward way of statistically testing for grouping around natural numbers, using estimated shape parameters from gamma duration fits, would be to subtract the nearest natural number from each of these estimates. This operation would produce a distribution of

*residual*shape parameters ranging from −0.5 to 0.5 that should be peaked around 0 in case of natural

*k*values. One can statistically test for the presence of such a peak using a standard test. We have performed such an analysis on our data without finding any evidence for natural shape parameters; however, Monte Carlo simulations show that a data set as large as ours (192 distributions of 243 points on average) does not provide enough statistical power to demonstrate natural

*k*values in this way, even if they are present. This is because the parameter estimates from any fitting procedure have only a limited accuracy, and we do not have enough data to detect a signal in this noise. Similar simulations show that Murata and colleagues’ data set (227 distributions of 350 data points) might be just large enough to successfully perform such an analysis, but it should be noted that these simulations were performed without adding any noise.

*rate*distribution. At present, we do not have any concrete neural model, but we might speculate that a perceptual alternation might occur as a threshold is reached by some decision signal that is itself the sum of a number of rising signals. If these rising signals would, in analogy to the classic Poisson model, have rates that are drawn from a Poisson distribution, perceptual alternation rates would follow gamma distributions, and these would have natural

*k*values. Clearly, this is only speculation, and Poisson distributed rates are less intuitive than Poisson distributed latencies, but we think mentioning this interpretation is worthwhile because of a particular characteristic of the gamma shape parameter: When a data set reasonably fits both a gamma duration distribution and a gamma rate distribution, the

*k*values of these two distributions are equal. This relation can be proven mathematically2, and we can confirm it for our fits (when we plotted the

*k*values from the fits to our 192 distributions against each other, linear regression gave

*k*

_{rate}= −0.02 + 1.02

*k*

_{duration}, with an

*r*

^{2}of 0.998). The important implication of this characteristic is that should one be able to support the Poisson clock model by finding the

*k*values of gamma duration distributions to take on natural values, this would automatically mean the same for a gamma rate model.

^{1}The gamma distribution and the beta′ distribution are linked by the fact that dividing two gamma distributions of equal scale parameter produces a beta′ distribution. The beta distribution (a scaled version of the beta′ distribution) previously made its appearance in bistable perception literature when Borsellino et al. (1972) made use of this feature in their analysis of supposedly gamma-distributed percept durations.

^{2}Based on the characteristic alluded to in the previous footnote, we can mathematically prove that the shape parameters from our gamma rate fits should be equal to those of our gamma duration fits. This can be understood as follows. If

*X*and

*Y*show gamma distributions with different shape parameters

*p*and

*q*, but equal scale parameter

*r*, then

*X*/

*Y*shows a beta′ distribution with parameters

*p*and

*q*(e.g., Pestman, 1998). Now by choosing for

*X*and

*Y*two halves of some set of percept durations that reasonably fits a gamma distribution, we make certain that

*p*=

*q*. Consequently, taking either

*X*/

*Y*or

*Y*/

*X*will both yield the same beta′ distribution with the two parameters equal. Clearly, if the same data set conforms to a gamma rate distribution, too, we might just as well divide 1/

*X*by 1/

*Y*to arrive at the same beta′ distribution. We can conclude that if some data set shows a fair fit to both a gamma rate distribution and a gamma duration distribution, both have an equal shape parameter.

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